Trigonometry/Angles of a triangle sum to 180 Degrees

From Wikibooks, open books for an open world
Jump to: navigation, search

In any triangle the angles
always sum to 180^{\circ}\,

The Sum of Angles[edit]

In any triangle the angles always sum to 180^{\circ}\,

This is a perhaps surprising fact.

Because 90^{\circ}\, is a right angle, it means that the sum of the angles of any triangle is the same as two right angles. If we 'tore the corners off' and placed them together at the same point, we could arrange them so that they exactly formed a straight line. There doesn't need to be anything special about the triangle. It works for any triangle.


Angles sum to 180o

Some examples that we had before of triangles are shown below

.
Equilateral Triangle 45-45-90 Triangle 30-60-90 Triangle 50-60-70 Triangle 20-40-120 Triangle
Equilateral Triangle 45-45-90 Triangle 30-60-90 Triangle 50-60-70 Triangle 20-40-120 Triangle
  • The first example shows an equilateral triangle. All of the sides are equal. All of the angles are equal. Each angle is 60 degrees. The sum of the angles is 60^{\circ}+60^{\circ}+60^{\circ}\, which is 180^{\circ}\,.
  • The second triangle shows a right angle triangle. One of the angles is a right angle. This right angle triangle has two sides the same length. It is symmetric. It fulfils our criteria for being an isosceles triangle. This is a particularly special isosceles triangle because it is isosceles and it is a right triangle. There is one angle of 90 degrees and each of the two remaining angles is 45 degrees. The sum of the angles is 45^{\circ}+45^{\circ}+90^{\circ}\, which is 180^{\circ}\,.
  • The third triangle is sometimes called the 30-60-90 triangle, because of its angles. It is actually half an equilateral triangle. The sum of the angles is 30^{\circ}+60^{\circ}+90^{\circ}\, which is 180^{\circ}\,.

The pattern is pretty clear.

  • Next we have a more arbitrary triangle. All the sides are different. The angles are 50, 60 and 70. The sum of the angles is 50^{\circ}+60^{\circ}+70^{\circ}\, which is 180^{\circ}\,.
  • Finally we have a triangle with an obtuse angle, that is one of the angles is larger than 90 degrees. The angles happen to be 20, 40 and 120, and the sum of the angles is 20^{\circ}+40^{\circ}+120^{\circ}\, which is 180^{\circ}\,.

The examples suggest it is true, but they don't prove it.[edit]

We could keep on doing this for other triangles, and keep finding the same answer, unless we make a mistake. This might convince us that our statement that the angles sum to 180 is true for all triangles, but it does not prove that it is so. To prove it we need some kind of general argument that could convince a mathematician that it is true. How do we know it is always true?

How could it go wrong? Well, if we hadn't tried with a triangle with an obtuse angle, it might be the case that the formula only works for triangles which don't have obtuse angles. Even having tried the triangle with an obtuse angle we could have not been trying hard enough to find an example that doesn't work. For all we know the formula only works if the angles are multiples of 5 degrees.

Proof will show it works for all triangles[edit]

The formula does in fact work for all triangles. We can for example make a triangle with angles of 33 degrees and 66 degrees and the third angle will have to be 81 degrees. Making more and more examples unfortunately doesn't get us anywhere closer to proving it is true of all triangles. We need a different approach. We'll show a proof later. The point of having a proof is to show that it is true for all triangles, not just the ones we've chosen to look at.

Exercises[edit]

A triangle has angles 30 degrees and 90 degrees, what is the third angle?
  •  
A triangle has angles 15 degrees and 65 degrees, what is the third angle?
  •  
A triangle has angles 100 degrees and 79.5 degrees, what is the third angle?
  •  
    • Do you think all the sides of this triangle will be about the same length?
What is the measure of each angle of an equilateral triangle?
  •  
Roadsign Exercise

The following road signs from Tanland show how steep the road ahead is. Put the road signs in order, least steep to steepest.

  • In these signs a sign that shows, for example, 5:8 means that the road is 5m higher when you've travelled 8m horizontally.
1:1
1:10
8:13
5:8
3:5
The measure of one angle of an isosceles triangle is \displaystyle 108^{\circ}. What are the measures of the other two angles?
  •